3.111 \(\int x^6 C(b x) \, dx\)

Optimal. Leaf size=109 \[ -\frac {x^6 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi b}+\frac {48 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^4 b^7}+\frac {24 x^2 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^3 b^5}-\frac {6 x^4 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^2 b^3}+\frac {1}{7} x^7 C(b x) \]

[Out]

48/7*cos(1/2*b^2*Pi*x^2)/b^7/Pi^4-6/7*x^4*cos(1/2*b^2*Pi*x^2)/b^3/Pi^2+1/7*x^7*FresnelC(b*x)+24/7*x^2*sin(1/2*
b^2*Pi*x^2)/b^5/Pi^3-1/7*x^6*sin(1/2*b^2*Pi*x^2)/b/Pi

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Rubi [A]  time = 0.11, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6427, 3380, 3296, 2638} \[ -\frac {x^6 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi b}+\frac {24 x^2 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^3 b^5}-\frac {6 x^4 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^2 b^3}+\frac {48 \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^4 b^7}+\frac {1}{7} x^7 \text {FresnelC}(b x) \]

Antiderivative was successfully verified.

[In]

Int[x^6*FresnelC[b*x],x]

[Out]

(48*Cos[(b^2*Pi*x^2)/2])/(7*b^7*Pi^4) - (6*x^4*Cos[(b^2*Pi*x^2)/2])/(7*b^3*Pi^2) + (x^7*FresnelC[b*x])/7 + (24
*x^2*Sin[(b^2*Pi*x^2)/2])/(7*b^5*Pi^3) - (x^6*Sin[(b^2*Pi*x^2)/2])/(7*b*Pi)

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3380

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Cos[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 6427

Int[FresnelC[(b_.)*(x_)]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*FresnelC[b*x])/(d*(m + 1)), x] -
Dist[b/(d*(m + 1)), Int[(d*x)^(m + 1)*Cos[(Pi*b^2*x^2)/2], x], x] /; FreeQ[{b, d, m}, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int x^6 C(b x) \, dx &=\frac {1}{7} x^7 C(b x)-\frac {1}{7} b \int x^7 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx\\ &=\frac {1}{7} x^7 C(b x)-\frac {1}{14} b \operatorname {Subst}\left (\int x^3 \cos \left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )\\ &=\frac {1}{7} x^7 C(b x)-\frac {x^6 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b \pi }+\frac {3 \operatorname {Subst}\left (\int x^2 \sin \left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{7 b \pi }\\ &=-\frac {6 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}+\frac {1}{7} x^7 C(b x)-\frac {x^6 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b \pi }+\frac {12 \operatorname {Subst}\left (\int x \cos \left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{7 b^3 \pi ^2}\\ &=-\frac {6 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}+\frac {1}{7} x^7 C(b x)+\frac {24 x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^5 \pi ^3}-\frac {x^6 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b \pi }-\frac {24 \operatorname {Subst}\left (\int \sin \left (\frac {1}{2} b^2 \pi x\right ) \, dx,x,x^2\right )}{7 b^5 \pi ^3}\\ &=\frac {48 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^7 \pi ^4}-\frac {6 x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^3 \pi ^2}+\frac {1}{7} x^7 C(b x)+\frac {24 x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b^5 \pi ^3}-\frac {x^6 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{7 b \pi }\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 83, normalized size = 0.76 \[ -\frac {6 \left (\pi ^2 b^4 x^4-8\right ) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^4 b^7}-\frac {x^2 \left (\pi ^2 b^4 x^4-24\right ) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{7 \pi ^3 b^5}+\frac {1}{7} x^7 C(b x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*FresnelC[b*x],x]

[Out]

(-6*(-8 + b^4*Pi^2*x^4)*Cos[(b^2*Pi*x^2)/2])/(7*b^7*Pi^4) + (x^7*FresnelC[b*x])/7 - (x^2*(-24 + b^4*Pi^2*x^4)*
Sin[(b^2*Pi*x^2)/2])/(7*b^5*Pi^3)

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fricas [F]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{6} {\rm fresnelc}\left (b x\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*fresnelc(b*x),x, algorithm="fricas")

[Out]

integral(x^6*fresnelc(b*x), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{6} {\rm fresnelc}\left (b x\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*fresnelc(b*x),x, algorithm="giac")

[Out]

integrate(x^6*fresnelc(b*x), x)

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maple [A]  time = 0.00, size = 107, normalized size = 0.98 \[ \frac {\frac {b^{7} x^{7} \FresnelC \left (b x \right )}{7}-\frac {b^{6} x^{6} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{7 \pi }+\frac {-\frac {6 b^{4} x^{4} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{7 \pi }+\frac {6 \left (\frac {4 b^{2} x^{2} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {8 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{7 \pi }}{\pi }}{b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*FresnelC(b*x),x)

[Out]

1/b^7*(1/7*b^7*x^7*FresnelC(b*x)-1/7/Pi*b^6*x^6*sin(1/2*b^2*Pi*x^2)+6/7/Pi*(-1/Pi*b^4*x^4*cos(1/2*b^2*Pi*x^2)+
4/Pi*(1/Pi*b^2*x^2*sin(1/2*b^2*Pi*x^2)+2/Pi^2*cos(1/2*b^2*Pi*x^2))))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{6} {\rm fresnelc}\left (b x\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*fresnelc(b*x),x, algorithm="maxima")

[Out]

integrate(x^6*fresnelc(b*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^6\,\mathrm {FresnelC}\left (b\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*FresnelC(b*x),x)

[Out]

int(x^6*FresnelC(b*x), x)

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sympy [A]  time = 2.23, size = 153, normalized size = 1.40 \[ \frac {x^{7} C\left (b x\right ) \Gamma \left (\frac {1}{4}\right )}{28 \Gamma \left (\frac {5}{4}\right )} - \frac {x^{6} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{28 \pi b \Gamma \left (\frac {5}{4}\right )} - \frac {3 x^{4} \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{14 \pi ^{2} b^{3} \Gamma \left (\frac {5}{4}\right )} + \frac {6 x^{2} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{7 \pi ^{3} b^{5} \Gamma \left (\frac {5}{4}\right )} + \frac {12 \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{7 \pi ^{4} b^{7} \Gamma \left (\frac {5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*fresnelc(b*x),x)

[Out]

x**7*fresnelc(b*x)*gamma(1/4)/(28*gamma(5/4)) - x**6*sin(pi*b**2*x**2/2)*gamma(1/4)/(28*pi*b*gamma(5/4)) - 3*x
**4*cos(pi*b**2*x**2/2)*gamma(1/4)/(14*pi**2*b**3*gamma(5/4)) + 6*x**2*sin(pi*b**2*x**2/2)*gamma(1/4)/(7*pi**3
*b**5*gamma(5/4)) + 12*cos(pi*b**2*x**2/2)*gamma(1/4)/(7*pi**4*b**7*gamma(5/4))

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